Q. To mop-clean a floor, a cleaning machine presses a circular mop of radius $R$ vertically down with a total force $F$ and rotates it with a constant angular speed about its axis. If the force $F$ is distributed uniformly over the mop and if coefficient of friction between the mop and the floor is $\mu$, the torque, applied by the machine on the mop is :
Solution:
Consider a strip of radius x & thickness dx, Torque due to friction on this strip.
$\int d\tau =\int^{R}_{0} \frac{x\mu F.2\pi x dx }{\pi R^{2}}
$
$\tau = \frac{ 2\mu F}{R^{2}} . \frac{R^{3}}{3} $
$ \tau = \frac{2 \mu FR}{ 3} $
